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Equation of State Michael Palmer
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Equations of State Common ones we’ve heard of Ideal gas Barotropic Adiabatic Fully degenerate Focus on fully degenerate EOS Specifically on corrections made to this EOS
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White Dwarfs Physics of WD’s provide a lot of opportunity to improve EOS of a fully degenerate gas Fully degenerate core Partially degenerate towards surface Crystallization (starting in the interior) Physics of crystallization
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Crystallization Interior of WD, ions can not be treated as ideal Must consider Coulomb interactions Electrons not effect at screening ions favourable for ions to rearrange themselves When 3kT/2 is of the order of -Ze 3kT/2 => Thermal energy -Ze => Coulomb energy per ion
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Crystallization Coulomb coupling ratio = (Ze) 2 / (r i kT) (Kippenhahn and Weigert, pg 134) –r i is the mean separation between ions »Defined as (3/(4 n i )) 1/3 »n i is the number density of ions –k is the Boltzmann constant and T is temperature Gives insight into the strength of the Coulomb interactions If >> 1, kinetic energy’s role not significant and ions will try and settle into a lower energy state Crystallization at ≈ 175 for one component plasma Intermediate values results in phase transition from a gas to a liquid
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Crystallization T m ≈ (Z 2 e 2 / c k)(4 /3 0 m u ) 1/3 critical temperature obtained from and density relation (Kippenhahn and Weigert, pg 134) Phase from liquid to solid can not be gradual Symmetry properties First order phase transition => lose latent heat Phase transition found to be first order (Winget et al. 2009) PUT IN FIGURE!!!! Latent heat ~ kT per ion, slows cooling Observed as bump => PUT IN FIGURE!!!!
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Crystallization Lattice ions oscillate Coulomb energy -E C = 2Z/(A 1/3 ) 6 1/3 keV »As T-> 0 ions not at rest Oscillate about points of equilibrium Frequency E 2 ~ Z 2 e 2 n 0 /m 0 ZE zp = 3 E h (bar) /2 and E zp = (0.6/A) 6 1/2 keV E zp << E C so does not contribute as much to E = E 0 + E C + E zp ~ E 0 + E C Find EC influences the pressure by lowering it as compared to an ideal Fermi gas –From P -dE / d(1/n)
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Crystallization Specific heat C v effect When << 1 the ions in the interior of the white dwarf behave as an ideal gas, C v = 3k/2 As ions form lattice, energy goes into lattice oscillations, results in additional degrees of freedom which raise C v to a maximum of 3k = 2c v MT / 5L
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Crystallization Electron polarization, Coulomb crystal f ie = -f ∞ (x r ) [1 + A(x r )(Q( )/ ) 8 ] »f ie is the correction to free energy »f ∞ (x r ) = b 1 √(1 + b 2 /x 2 ) »A(x r ) = (b 3 + a 3 x 2 ) / (1 + b 4 x 2 ) »Q( ) classically defined as ≈ q » is T P /T »xr = p F /m e c »b1,b2,b3,b4,a3 are all constants »q = 0.205 Form of Q( ) is redefined as Q( ) = [ln(1 + e (qh) 2 )] 1/2 [ln(e - (e - 2)e -(qh) 2 )] -1/2
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PC EOS & MESA Low temperature high density region Carefully handles mixtures of carbon and oxygen Accounts for all corrections to EOS laid out earlier and more Ex => Inverse Beta Decay Can handle both classical and quantum Coulomb crystals, Coulomb liquid interactions (weak or strong coupling),… Default for > 80
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